Group #45665935 — subgroups

Subgroups — #45665935

Order 32896.

228 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22551598	C₂	1	1	yes	maximal, characteristic
2	C₂²×#16861593	C₂	1	1	yes	maximal, characteristic, complemented
3	#30656516	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16861618	C₄	1	1	yes	characteristic
6	#22551598	—	2	2	no	—
7	#22551598	—	2	2	no	—
8	C₂×#16861602	C₄	1	1	yes	characteristic
9	C₂×#16861601	C₄	1	2	yes	—
10	C₂×C₈×D₂₅₇	C₂²	1	2	yes	—
11	C₂×#16861593	—	2	2	no	—
12	C₂×#16861593	C₂²	1	2	yes	—
13	C₂²×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
14	M₃₂×C₂₅₇	—	2	2	no	—
15	C₂×C₁₆×C₂₅₇	C₂²	1	2	yes	—
16	#16861618	—	2	2	no	—
17	#16861618	—	2	2	no	—
18	C₂×Dic₅₁₄	C₈	1	1	yes	characteristic
19	C₂×D₁₀₂₈	C₈	1	1	yes	characteristic
20	C₂×C₂²⋊D₂₅₇	C₈	1	2	yes	—
21	C₂×C₄×D₂₅₇	C₂×C₄	1	2	yes	—
22	#16861602	—	2	2	no	—
23	#16861601	—	2	4	no	—
24	C₂×#14523339	C₂×C₄	1	2	yes	—
25	C₈×D₂₅₇	—	2	2	no	—
26	C₈×D₂₅₇	—	2	2	no	—
27	#16861593	—	2	2	no	—
28	C₂²×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₂₅₇	D₄	1	2	yes	—
31	C₂×C₈×C₂₅₇	D₄	1	2	yes	—
32	C₂×C₈×C₂₅₇	C₂³	1	1	yes	characteristic
33	C₁₆×C₂₅₇	—	2	2	no	—
34	C₂²×D₂₅₇	C₂×C₈	1	2	yes	—
35	Dic₅₁₄	—	2	2	no	—
36	D₁₀₂₈	—	2	2	no	—
37	C₂²⋊D₂₅₇	—	2	4	no	—
38	C₂×Dic₂₅₇	C₂×C₈	1	2	yes	—
39	C₄×D₂₅₇	—	2	2	no	—
40	C₄×D₂₅₇	—	2	2	no	—
41	#14523339	—	2	2	no	—
42	C₂³×C₂₅₇	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₂₅₇	C₂×C₈	1	1	yes	characteristic
45	C₂×C₄×C₂₅₇	M₁₆	1	2	yes	—
46	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
47	C₈×C₂₅₇	C₂²⋊C₄	1	2	yes	—
48	C₈×C₂₅₇	C₂×D₄	1	1	yes	characteristic
49	C₈×C₂₅₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₂₅₇	—	2	2	no	—
51	C₂²×C₂₅₇	C₂²⋊C₈	1	2	yes	—
52	C₂²×C₂₅₇	C₂²×C₈	1	1	yes	characteristic
53	C₂×D₂₅₇	—	2	2	no	—
54	C₂×D₂₅₇	—	2	4	no	—
55	C₄×C₂₅₇	C₂²⋊C₈	1	2	yes	—
56	C₄×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₄×C₂₅₇	C₂×M₁₆	1	1	yes	characteristic
58	Dic₂₅₇	—	2	2	no	—
59	C₂×C₂₅₇	—	2	2	no	—
60	C₂×C₂₅₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
61	C₂×C₂₅₇	#504	1	2	yes	—
62	D₂₅₇	—	4	2	no	—
63	C₂₅₇	#2113	1	1	yes	characteristic, complemented, Sylow
64	#2113	—	257	1	no	maximal, Sylow
65	C₂×#131	—	257	1	no	—
66	C₂×M₃₂	D₂₅₇	1	1	yes	characteristic, complemented
67	C₂²×C₁₆	—	257	1	no	—
68	C₂²⋊C₁₆	—	257	4	no	—
69	C₂×#41	—	257	1	no	—
70	#131	—	514	2	no	—
71	#131	—	514	2	no	—
72	C₂×M₁₆	—	257	1	no	—
73	C₂×M₁₆	—	257	2	no	—
74	C₂²×C₈	—	257	2	no	—
75	C₂²×C₈	C₂×D₂₅₇	1	1	yes	characteristic
76	M₃₂	—	2	2	no	—
77	C₂×C₁₆	—	257	2	no	—
78	C₂×C₁₆	—	514	2	no	—
79	C₂×C₁₆	C₂×D₂₅₇	1	2	yes	—
80	#41	—	514	2	no	—
81	#41	—	514	2	no	—
82	C₂×Q₈	—	257	1	no	—
83	C₂×D₄	—	257	2	no	—
84	C₂×D₄	—	257	1	no	—
85	C₂²×C₄	—	257	2	no	—
86	C₂²×C₄	C₄×D₂₅₇	1	1	yes	characteristic
87	M₁₆	—	514	2	no	—
88	M₁₆	—	514	4	no	—
89	C₂×C₈	—	257	2	no	—
90	C₂×C₈	—	514	2	no	—
91	C₂×C₈	—	514	2	no	—
92	C₂×C₈	C₄×D₂₅₇	1	1	yes	characteristic
93	C₂×C₈	C₂²⋊D₂₅₇	1	2	yes	—
94	C₂×C₈	D₁₀₂₈	1	2	yes	—
95	C₂×C₈	C₂²×D₂₅₇	1	1	yes	characteristic
96	C₁₆	—	2	2	no	—
97	C₁₆	—	514	2	no	—
98	C₂³	—	257	2	no	—
99	C₂³	C₈×D₂₅₇	1	1	yes	characteristic
100	Q₈	—	514	2	no	—
101	D₄	—	514	4	no	—
102	D₄	—	514	2	no	—
103	C₂×C₄	—	257	2	no	—
104	C₂×C₄	—	514	2	no	—
105	C₂×C₄	—	514	2	no	—
106	C₂×C₄	#16861595	1	2	yes	—
107	C₂×C₄	C₈×D₂₅₇	1	1	yes	characteristic
108	C₂×C₄	#16861601	1	2	yes	—
109	C₂×C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
110	C₈	—	514	2	no	—
111	C₈	#16861595	1	2	yes	—
112	C₈	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
113	C₈	C₂×D₁₀₂₈	1	1	yes	characteristic
114	C₂²	—	2	2	no	—
115	C₂²	—	514	4	no	—
116	C₂²	—	514	2	no	—
117	C₂²	#22551504	1	2	yes	—
118	C₂²	C₂×C₈×D₂₅₇	1	1	yes	characteristic
119	C₄	—	514	2	no	—
120	C₄	#22551504	1	2	yes	—
121	C₄	C₂×#16861595	1	1	yes	characteristic
122	C₄	C₂×#16861601	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	1028	2	no	—
125	C₂	C₂×#22551504	1	1	yes	characteristic
126	C₂	#30657155	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22551598

C₂

yes

maximal, characteristic

C₂²×#16861593

C₂

yes

maximal, characteristic, complemented

#30656516

C₂

yes

maximal, complemented

C₂×M₃₂×C₂₅₇

C₂

yes

maximal, characteristic, complemented

C₂×#16861618

C₄

yes

—

—

C₄

yes

characteristic

C₂×#16861601

C₄

yes

—

C₂×C₈×D₂₅₇

C₂²

yes

—

C₂×#16861593

—

C₂×#16861593

C₂²

yes

—

C₂²×C₈×C₂₅₇

C₂²

yes

characteristic

M₃₂×C₂₅₇

—

C₂×C₁₆×C₂₅₇

C₂²

yes

—

#16861618

—

#16861618

—

C₂×Dic₅₁₄

C₈

yes

characteristic

C₂×D₁₀₂₈

C₈

yes

characteristic

C₂×C₂²⋊D₂₅₇

C₈

yes

—

C₂×C₄×D₂₅₇

C₂×C₄

yes

—

#16861602

—

#16861601

—

C₂×#14523339

C₂×C₄

yes

—

C₈×D₂₅₇

—

C₈×D₂₅₇

—

#16861593

—

C₂²×C₄×C₂₅₇

C₂×C₄

yes

characteristic

C₂×C₈×C₂₅₇

C₂×C₄

yes

characteristic

C₂×C₈×C₂₅₇

D₄

yes

—

C₂×C₈×C₂₅₇

D₄

yes

—

C₂×C₈×C₂₅₇

C₂³

yes

characteristic

C₁₆×C₂₅₇

—

C₂²×D₂₅₇

C₂×C₈

yes

—

Dic₅₁₄

—

D₁₀₂₈

—

C₂²⋊D₂₅₇

—

C₂×Dic₂₅₇

C₂×C₈

yes

—

C₄×D₂₅₇

—

C₄×D₂₅₇

—

#14523339

—

C₂³×C₂₅₇

C₂×C₈

yes

characteristic

C₂×C₄×C₂₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₇

C₂×C₈

yes

characteristic

C₂×C₄×C₂₅₇

M₁₆

yes

—

C₂×C₄×C₂₅₇

C₂²×C₄

yes

characteristic

C₈×C₂₅₇

C₂²⋊C₄

yes

—

C₈×C₂₅₇

C₂×D₄

yes

characteristic

C₈×C₂₅₇

C₂×D₄

yes

characteristic

C₂²×C₂₅₇

—

C₂²×C₂₅₇

C₂²⋊C₈

yes

—

C₂²×C₂₅₇

C₂²×C₈

yes

characteristic

C₂×D₂₅₇

—

C₂×D₂₅₇

—

C₄×C₂₅₇

C₂²⋊C₈

yes

—

C₄×C₂₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₅₇

C₂×M₁₆

yes

characteristic

Dic₂₅₇

—

C₂×C₂₅₇

—

C₂×C₂₅₇

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₂₅₇

#504

yes

—

D₂₅₇

—

C₂₅₇

#2113

yes

characteristic, complemented, Sylow

#2113

—

257

maximal, Sylow

C₂×#131

—

257

—

C₂×M₃₂

D₂₅₇

yes

characteristic, complemented

C₂²×C₁₆

—

257

—

C₂²⋊C₁₆

—

257

—

C₂×#41

—

257

—

#131

—

514

—

#131

—

514

—

C₂×M₁₆

—

257

—

C₂×M₁₆

—

257

—

C₂²×C₈

—

257

—

C₂²×C₈

C₂×D₂₅₇

yes

characteristic

M₃₂

—

C₂×C₁₆

—

257

—

C₂×C₁₆

—

514

—

C₂×C₁₆

C₂×D₂₅₇

yes

—

#41

—

514

—

#41

—

514

—

C₂×Q₈

—

257

—

C₂×D₄

—

257

—

C₂×D₄

—

257

—

C₂²×C₄

—

257

—

C₂²×C₄

C₄×D₂₅₇

yes

characteristic

M₁₆

—

514

—

M₁₆

—

514

—

C₂×C₈

—

257

—

C₂×C₈

—

514

—

C₂×C₈

—

514

—

C₂×C₈

C₄×D₂₅₇

yes

characteristic

C₂×C₈

C₂²⋊D₂₅₇

yes

—

C₂×C₈

D₁₀₂₈

yes

—

C₂×C₈

C₂²×D₂₅₇

yes

characteristic

C₁₆

—

C₁₆

—

514

—

C₂³

—

257

—

C₂³

C₈×D₂₅₇

yes

characteristic

100

Q₈

—

514

—

101

D₄

—

514

—

102

D₄

—

514

—

103

C₂×C₄

—

257

—

104

C₂×C₄

—

514

—

105

C₂×C₄

—

514

—

106

C₂×C₄

#16861595

yes

—

107

C₂×C₄

C₈×D₂₅₇

yes

characteristic

108

C₂×C₄

#16861601

yes

—

109

C₂×C₄

C₂×C₄×D₂₅₇

yes

characteristic

110

C₈

—

514

—

111

C₈

#16861595

yes

—

112

C₈

C₂×C₂²⋊D₂₅₇

yes

characteristic

113

C₈

C₂×D₁₀₂₈

yes

characteristic

114

C₂²

—

115

C₂²

—

514

—

116

C₂²

—

514

—

117

C₂²

#22551504

yes

—

118

C₂²

C₂×C₈×D₂₅₇

yes

characteristic

119

C₄

—

514

—

120

C₄

#22551504

yes

—

121

C₄

C₂×#16861595

yes

characteristic

122

C₄

C₂×#16861601

yes

characteristic

123

C₂

—

124

C₂

—

1028

—

125

C₂

C₂×#22551504

yes

characteristic

126

C₂

#30657155

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₂₅₇
commutator, socle	C₂²×C₂₅₇	C₂²×C₈
Fitting	C₂×M₃₂×C₂₅₇	C₂
prime radical	C₂×#16861618	C₄
cosocle	C₂×C₈×C₂₅₇	C₂³
hypercenter	C₂×M₃₂	D₂₅₇
hypocenter	C₂₅₇	#2113
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₂₅₇

commutator, socle

C₂²×C₂₅₇

C₂²×C₈

Fitting

C₂×M₃₂×C₂₅₇

C₂

prime radical

C₂×#16861618

C₄

cosocle

C₂×C₈×C₂₅₇

C₂³

hypercenter

C₂×M₃₂

D₂₅₇

C₂₅₇

G ▷ C₂²×C₂₅₇ ▷ C₂₅₇

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
2. C₂²×#16861593	C₂	semidirect
3. #30656516	C₂	semidirect
4. C₂×M₃₂×C₂₅₇	C₂	semidirect
63. C₂₅₇	#2113	semidirect
66. C₂×M₃₂	D₂₅₇	semidirect

H ≅ G/N

type

2. C₂²×#16861593

C₂

semidirect

3. #30656516

C₂

semidirect

4. C₂×M₃₂×C₂₅₇

C₂

semidirect

63. C₂₅₇

#2113

semidirect

66. C₂×M₃₂

D₂₅₇

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#22551598	2
C₂²×#16861593	2
#30656516	2
C₂×M₃₂×C₂₅₇	2
#2113	257

subgroup

index

C₂×M₃₂×C₂₅₇

257

Subgroups — #45665935

Conjugacy classes of subgroups

Complements

Maximal subgroups